Calculus Examples

$f (x) = 4 x^{3} - \frac{12}{x^{3}} + \frac{x^{2}}{5} + \frac{2}{\sqrt{x}} - \frac{4}{4 \sqrt{x^{3}}}$

Step 1

By the Sum Rule, the derivative of $4 x^{3} - \frac{12}{x^{3}} + \frac{x^{2}}{5} + \frac{2}{\sqrt{x}} - \frac{4}{4 \sqrt{x^{3}}}$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- \frac{12}{x^{3}}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- \frac{12}{x^{3}}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{12}{x^{3}}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [- \frac{12}{x^{3}}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- \frac{12}{x^{3}}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{12}{x^{3}}]$ .

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Step 3.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- \frac{12}{x^{3}}$ with respect to $x$ is $- 12 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$12 x^{2} - 12 \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$12 x^{2} - 12 \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$12 x^{2} - 12 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = - 1$ .

$12 x^{2} - 12 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$12 x^{2} - 12 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$12 x^{2} - 12 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 3.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$12 x^{2} - 12 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.5.2

Multiply $3$ by $- 2$ .

$12 x^{2} - 12 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.6

Multiply $3$ by $- 1$ .

$12 x^{2} - 12 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.7.1

Move $x^{2}$ .

$12 x^{2} - 12 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.7.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{2} - 12 (- 3 x^{2 - 6}) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.7.3

Subtract $6$ from $2$ .

$12 x^{2} - 12 (- 3 x^{- 4}) + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 3.8

Multiply $- 3$ by $- 12$ .

$12 x^{2} + 36 x^{- 4} + \frac{d}{d x} [\frac{x^{2}}{5}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 4

Evaluate $\frac{d}{d x} [\frac{x^{2}}{5}]$ .

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Step 4.1

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{2}]$ .

$12 x^{2} + 36 x^{- 4} + \frac{1}{5} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$12 x^{2} + 36 x^{- 4} + \frac{1}{5} (2 x) + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 4.3

Combine $2$ and $\frac{1}{5}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2}{5} x + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 4.4

Combine $\frac{2}{5}$ and $x$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{d}{d x} [\frac{2}{\sqrt{x}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5

Evaluate $\frac{d}{d x} [\frac{2}{\sqrt{x}}]$ .

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{d}{d x} [\frac{2}{x^{\frac{1}{2}}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x^{\frac{1}{2}}}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}]$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 \frac{d}{d x} [\frac{1}{x^{\frac{1}{2}}}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.3

Rewrite $\frac{1}{x^{\frac{1}{2}}}$ as ${(x^{\frac{1}{2}})}^{- 1}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 \frac{d}{d x} [{(x^{\frac{1}{2}})}^{- 1}] + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 5.4.1

To apply the Chain Rule, set $u_{2}$ as $x^{\frac{1}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = - 1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.4.3

Replace all occurrences of $u_{2}$ with $x^{\frac{1}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- {(x^{\frac{1}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{1}{2}}]) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- {(x^{\frac{1}{2}})}^{- 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 5.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{\frac{1}{2} \cdot - 2} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.6.2

Cancel the common factor of $2$ .

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Step 5.6.2.1

Factor $2$ out of $- 2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.6.2.2

Cancel the common factor.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.6.2.3

Rewrite the expression.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.8

Combine $- 1$ and $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.9

Combine the numerators over the common denominator.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.10

Simplify the numerator.

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Step 5.10.1

Multiply $- 1$ by $2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{\frac{1 - 2}{2}})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.10.2

Subtract $2$ from $1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{\frac{- 1}{2}})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.11

Move the negative in front of the fraction.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} (\frac{1}{2} x^{- \frac{1}{2}})) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.12

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- x^{- 1} \frac{x^{- \frac{1}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.13

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $x^{- 1}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{- \frac{1}{2}} x^{- 1}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 5.14.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{- \frac{1}{2} - 1}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14.3

Combine $- 1$ and $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14.4

Combine the numerators over the common denominator.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{\frac{- 1 - 1 \cdot 2}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14.5

Simplify the numerator.

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Step 5.14.5.1

Multiply $- 1$ by $2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{\frac{- 1 - 2}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14.5.2

Subtract $2$ from $- 1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{\frac{- 3}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.14.6

Move the negative in front of the fraction.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{x^{- \frac{3}{2}}}{2}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.15

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + 2 (- \frac{1}{2 x^{\frac{3}{2}}}) + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.16

Multiply $- 1$ by $2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - 2 \frac{1}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.17

Combine $- 2$ and $\frac{1}{2 x^{\frac{3}{2}}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{- 2}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.18

Factor $2$ out of $- 2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{2 \cdot - 1}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.19

Cancel the common factors.

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Step 5.19.1

Factor $2$ out of $2 x^{\frac{3}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{2 \cdot - 1}{2 (x^{\frac{3}{2}})} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.19.2

Cancel the common factor.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{2 \cdot - 1}{2 x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.19.3

Rewrite the expression.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} + \frac{- 1}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 5.20

Move the negative in front of the fraction.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 6

Evaluate $\frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$ .

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Step 6.1

Cancel the common factor of $4$ .

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Step 6.1.1

Cancel the common factor.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{4}{4 \sqrt{x^{3}}}]$

Step 6.1.2

Rewrite the expression.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{1}{\sqrt{x^{3}}}]$

Step 6.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{d}{d x} [- \frac{1}{x^{\frac{3}{2}}}]$

Step 6.3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{\frac{3}{2}}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - \frac{d}{d x} [\frac{1}{x^{\frac{3}{2}}}] + \frac{1}{x^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

Step 6.4

Rewrite $\frac{1}{x^{\frac{3}{2}}}$ as ${(x^{\frac{3}{2}})}^{- 1}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - \frac{d}{d x} [{(x^{\frac{3}{2}})}^{- 1}] + \frac{1}{x^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

Step 6.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{\frac{3}{2}}$ .

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Step 6.5.1

To apply the Chain Rule, set $u_{3}$ as $x^{\frac{3}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (\frac{d}{d u_{3}} [{u_{3}}^{- 1}] \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{1}{x^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

Step 6.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{3}} [{u_{3}}^{n}]$ is $n {u_{3}}^{n - 1}$ where $n = - 1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- {u_{3}}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{1}{x^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

Step 6.5.3

Replace all occurrences of $u_{3}$ with $x^{\frac{3}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- {(x^{\frac{3}{2}})}^{- 2} \frac{d}{d x} [x^{\frac{3}{2}}]) + \frac{1}{x^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

Step 6.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \frac{d}{d x} [- 1]$

Step 6.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- {(x^{\frac{3}{2}})}^{- 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.8

Multiply the exponents in ${(x^{\frac{3}{2}})}^{- 2}$ .

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Step 6.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{\frac{3}{2} \cdot - 2} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.8.2

Cancel the common factor of $2$ .

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Step 6.8.2.1

Factor $2$ out of $- 2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{\frac{3}{2} \cdot (2 (- 1))} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.8.2.2

Cancel the common factor.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{\frac{3}{2} \cdot (2 \cdot - 1)} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.8.2.3

Rewrite the expression.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{3 \cdot - 1} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.8.3

Multiply $3$ by $- 1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.9

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.10

Combine $- 1$ and $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.11

Combine the numerators over the common denominator.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.12

Simplify the numerator.

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Step 6.12.1

Multiply $- 1$ by $2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} (\frac{3}{2} x^{\frac{3 - 2}{2}})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.12.2

Subtract $2$ from $3$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} (\frac{3}{2} x^{\frac{1}{2}})) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.13

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - (- x^{- 3} \frac{3 x^{\frac{1}{2}}}{2}) + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.14

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $x^{- 3}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{\frac{1}{2}} x^{- 3}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15

Multiply $x^{\frac{1}{2}}$ by $x^{- 3}$ by adding the exponents.

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Step 6.15.1

Move $x^{- 3}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 (x^{- 3} x^{\frac{1}{2}})}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{- 3 + \frac{1}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.3

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{- 3 \cdot \frac{2}{2} + \frac{1}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.4

Combine $- 3$ and $\frac{2}{2}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{\frac{- 3 \cdot 2}{2} + \frac{1}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.5

Combine the numerators over the common denominator.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{\frac{- 3 \cdot 2 + 1}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.6

Simplify the numerator.

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Step 6.15.6.1

Multiply $- 3$ by $2$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{\frac{- 6 + 1}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.6.2

Add $- 6$ and $1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{\frac{- 5}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.15.7

Move the negative in front of the fraction.

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3 x^{- \frac{5}{2}}}{2} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.16

Move $x^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} - - \frac{3}{2 x^{\frac{5}{2}}} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.17

Multiply $- 1$ by $- 1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + 1 \frac{3}{2 x^{\frac{5}{2}}} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.18

Multiply $\frac{3}{2 x^{\frac{5}{2}}}$ by $1$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{3}{2 x^{\frac{5}{2}}} + \frac{1}{x^{\frac{3}{2}}} \cdot 0$

Step 6.19

Multiply $\frac{1}{x^{\frac{3}{2}}}$ by $0$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{3}{2 x^{\frac{5}{2}}} + 0$

Step 6.20

Add $\frac{3}{2 x^{\frac{5}{2}}}$ and $0$ .

$12 x^{2} + 36 x^{- 4} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{3}{2 x^{\frac{5}{2}}}$

Step 7

Simplify.

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Step 7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$12 x^{2} + 36 \frac{1}{x^{4}} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{3}{2 x^{\frac{5}{2}}}$

Step 7.2

Combine $36$ and $\frac{1}{x^{4}}$ .

$12 x^{2} + \frac{36}{x^{4}} + \frac{2 x}{5} - \frac{1}{x^{\frac{3}{2}}} + \frac{3}{2 x^{\frac{5}{2}}}$

Step 7.3

Reorder terms.

$12 x^{2} + \frac{2 x}{5} + \frac{36}{x^{4}} + \frac{3}{2 x^{\frac{5}{2}}} - \frac{1}{x^{\frac{3}{2}}}$